Householder, Givens, and the QR factorization

Reading: Golub & Van Loan §5.1–5.2, pp. 234–260.

By the end of this week you should be able to

• Construct Householder reflections and Givens rotations and apply them stably.
• Compute a QR factorization by Householder triangularization.
• Explain why classical Gram-Schmidt loses orthogonality and what to do instead.

Algorithms introduced

• Householder QR
• Givens rotations
• Modified Gram-Schmidt

Where this shows up in AI

Orthogonal parameterizations keep recurrent and deep networks from exploding or vanishing; QR is how an orthonormal basis is built safely.

Materials